where \({R}\)  is the scalar curvature, \(\Phi \)  is the dilaton field,
\(F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }\) 
is the electromagnetic field tensor, and \(A_{\mu }\)  is the
electromagnetic potential. \(\alpha \)  is an arbitrary constant
governing the strength of the coupling between the dilaton and the
Maxwell field. The last term in Eq. (REF ) is the
Gibbons-Hawking surface term. It is required for the variational
principle to be well-defined. The factor \(\Theta \)  represents the
trace of the extrinsic curvature for the boundary \({\partial \mathcal {M}}\)  and \(\gamma \)  is the induced metric on the boundary.
While \(\alpha =0\)  corresponds to the usual Einstein-Maxwell-scalar
theory, \(\alpha =1\)  indicates the dilaton-electromagnetic coupling
that appears in the low energy string action in Einstein's frame.
For arbitrary value of \(\alpha \)  in AdS space the form of the
dilaton potential \(V(\Phi )\)  is chosen as [1]}
\(V(\Phi )=\frac{2\Lambda }{3(\alpha ^2+1)^{2}}\left[ {\alpha }^{2}\left( 3\,{\alpha }^{2}-1 \right) {e ^{-2\Phi /\alpha }}+ \left(3-{\alpha }^{2} \right) {e^{2\,\alpha \,\Phi }}+8\,{\alpha }^{2}{e^{\Phi (\alpha -1/\alpha )}} \right]. \) 
